Algorithm extends Euclidean cell decomposition to projective surfaces.
problem Computing Euclidean cell decomposition for non-hyperbolic surfaces.
method Generalised Weeks' algorithm to strictly convex projective surfaces.
result Algorithm successfully decomposes Euclidean cell structure for projective surfaces.
The moduli space of projective structures is tessellated using Euclidean cell decompositions.
problem Tessellating the moduli space of strictly convex projective structures.
method Using Euclidean cell decompositions, coordinates from Fock and Goncharov, and mapping class group actions.
result The moduli space has a natural cell decomposition.
The paper extends Delaunay decompositions to higher signature spaces.
problem Generalizing Delaunay decompositions to non-Euclidean spaces.
method Introducing a new family of regions bounded by quadratic hypersurfaces.
result Existence and uniqueness of the generalized Delaunay decomposition.
We study the moduli space of euclidean structures with cone points on a surface, and describe a decomposition into cells each of which corresponds to a given combinatorial type of Delaunay tessellation. We use some of the ideas to study hyperbolic structures on three-dimensional manifolds
Constructs a cell decomposition for the Fulton MacPherson operad FM_2.
problem Cellular decomposition of the Fulton MacPherson operad FM_2.
method Indexed by trees with two colors and cacti operad cells, compatible with operad composition.
result Computes generating functions for cell counts, algebraic.
We extend cell decomposition to moduli space of convex projective structures.
problem Cell decomposition of moduli space of convex projective structures.
method Use Fock and Goncharov's A-coordinates and edge-flipping algorithm. result Holonomy groups are semi-arithmetic in many cases.
The paper explores constructing an invariant for s-move 3-cells using 2-cell decompositions.
problem Creating an invariant for s-move 3-cells.
method Using elementary 3-expansions and 2-cell decompositions, the paper constructs an invariant.
result The method provides a sequence of 2-cells to decompose s-move 3-cells.
Model place cells as spatial embeddings for efficient path planning and cognitive map construction.
problem Encoding spatial navigation in the hippocampus.
method Model place cells using spectral decomposition of multi-step random walk transition kernels, inducing sparsity and adjacency.
result Place cells encode spatial information through non-negativity and inner-product structure, forming a cognitive map.
The moduli space of Riemann surfaces with at least two punctures can be decomposed into a cell complex by using a particular family of ribbon graphs called Nakamura graphs. We distinguish the moduli space with all punctures labelled from that with a single labelled puncture. In both cases, we describe a cell decomposit…
The study describes a cell structure for multisets in a rectangle.
problem Understanding the space of multisets in a rectangle.
method Developed a piecewise Euclidean bi-simplicial cell structure.
result Connected to spaces of complex polynomials and permutahedra.
Constructs a simplicial cell decomposition of complex projective space for n ≥ 2.
problem Finding a simplicial cell decomposition for complex projective space.
method Starting with a standard crystallisation of the 2-sphere, constructing a simplicial subdivision, and quotienting by the Sym(n) action.
result Explicit construction of a simplicial cell decomposition of complex projective space for n ≥ 2.
A Delaunay cell decomposition of a surface with constant curvature gives rise to a circle pattern, consisting of the circles which are circumscribed to the facets. We treat the problem whether there exists a Delaunay cell decomposition for a given (topological) cell decomposition and given intersection angles of the ci…
This survey covers earlier work of the author as well as recent work on Riemann's moduli space, its canonical cell decomposition and compactification, and the related operadic structure of arc complexes.
In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes which come equipped with natural integer points and a volume form. …
Euler's theorem extended to complex structures.
problem Generalizing Euler's theorem to complex structures.
method Analyzing strongly connected, pure n-dimensional regular CW-complexes. result Evenness of cells is equivalent to generalized cycle decomposition and traversability.
Quantization on even-dimensional compact manifolds using cell decomposition.
problem Quantization of compact even-dimensional manifolds.
method Cell decomposition and embedding in CP^d, inducing local Poisson structure and star product.
result Achieved Berezin-type quantization on compact even-dimensional manifolds.
A finite subset S of a closed hyperbolic surface F canonically determines a "centered dual decomposition" of F: a cell structure with vertex set S, geodesic edges, and 2-cells that are unions of the corresponding Delaunay polygons. Unlike a Delaunay polygon, a centered dual 2-cell Q is not determined by its collection …
Paper introduces signal processing on cell complexes.
problem Processing signals on non-Euclidean domains.
method Signal processing on abstract regular cell complexes.
result Hodge Laplacians for cell complexes enable convolutional filters.
The mapping class group invariant ideal cell decomposition of the Teichmueller space of a punctured surface times an open simplex has been used in a number of computations. This paper answers a question about the asymptotics of this decomposition, namely, in a given cell of the decomposition, which curves can be short?…
A method decomposes battery cell capacity trends using MCGP for high accuracy and uncertainty.
problem Forecasting lithium-ion battery cells capacity with high accuracy and uncertainty.
method Multi-Output Convolved Gaussian Process (MCGP) for latent function decomposition.
result The MCGP method provides high prediction accuracy and uncertainty information.
Decomposition theory explores topological spaces and their quotient spaces.
problem Understanding the topology of quotient spaces given a decomposition.
method Analyzing upper semi-continuous decompositions and their shrinkability.
result An upper semi-continuous decomposition yields a homeomorphic quotient space under certain conditions.
Researchers decompose hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
problem Decomposing hyperbolic n-manifolds with totally geodesic boundaries into polyhedral cells.
method Two different approaches to demonstrate the existence of polyhedral decompositions.
result The number of polyhedral decompositions of M is finite. In this note, we introduce a class of cell decompositions of PL manifolds and polyhedra which are more general than triangulations yet not as general as CW complexes; we propose calling them PLCW complexes. The main result is an analog of Alexander's theorem: any two PLCW decompositions of the same polyhedron can be ob…
In this paper, we study face vectors of simplicial posets that are the face posets of cell decompositions of topological manifolds without boundary. We characterize all possible face vectors of simplicial posets whose geometric realizations are homeomorphic to the product of spheres. As a corollary, we obtain the chara…
We visualize hyperbolic honeycombs using Schläfli symbols.
problem Tiling hyperbolic spaces with efficient Schläfli symbols.
method Develop strategies for visualizing honeycombs in spherical, Euclidean, and hyperbolic spaces.
result Infinitely many hyperbolic honeycombs exist, with various categories of vertices and cells.
Topological proof of Weil-Petersson symplectic form using Fenchel-Nielsen coordinates.
problem Proving Wolpert's formula for the Weil-Petersson symplectic form.
method Introducing a cell decomposition and groupoid cocycle on a surface to represent points in Teichmüller space.
result Topological proof of Wolpert's formula for the Weil-Petersson symplectic form.
A family of coordinates ψh for the Teichmüller space of a compact surface with boundary was introduced in \cite{l2}. In the work \cite{m1}, Mondello showed that the coordinate ψ0 can be used to produce a natural cell decomposition of the Teichmüller space invariant under the action of the mapping class group. In …
Enhances forecasting of complex systems using FKMD.
problem Forecasting high-dimensional dynamical systems with unknown features.
method Featurized Koopman Mode Decomposition (FKMD) using delay embedding and learned Mahalanobis distance.
result Improves prediction accuracy for various complex systems.
Study pentagon growth with laser-cut models.
problem Explore topological and geometric properties of pentagon cell growth.
method Cell growth process in Euclidean plane, physical representations created with laser cutter.
result Aesthetic and geometric insights from pentagon growth models.
Proves least Gaussian perimeter decomposition conjectures for 2-3 cells in n-dimensional space.
problem Finding least perimeter ways to divide space into cells of prescribed Gaussian measure.
method Analyzes stable clusters and uses Voronoi cells of equidistant points.
result Simplicial clusters are unique minimizers for 2-3 cells in n-dimensional space.
CW decomposition of manifolds with Morse functions.
problem CW decomposition of manifolds with Morse functions.
method Generic gradientlike vector field and Morse function.
result Stable manifolds provide a CW decomposition.
Proposes CXNs for neural network computations on cell complexes.
problem Performing neural network computations on complex topological spaces.
method Introduces a message passing scheme and a unified encoder-decoder framework for cell complexes.
result Generalizes message passing to cell complexes and provides a cell2vec representation.
The paper develops a theory of discrete Riemann surfaces using quadrilateral cells.
problem Developing a theory for discrete Riemann surfaces.
method Quadrilateral cellular decompositions and complex weights.
result New notions and results including branched coverings, discrete Riemann-Hurwitz Formula, and Abel-Jacobi map.
In 1998, Goresky, Kottwitz, and MacPherson showed that for certain spaces X equipped with a torus action, the T-equivariant cohomology ring of X can be described by combinatorial data obtained from its orbit decomposition. Thus, their theory transforms calculations of the equivariant topology of X to those of the combi…
Proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
problem Existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
method Construct an isotopic map instead of edge-flipping algorithm, generalizing Dyer et al's method.
result Strict proof of existence and uniqueness of weighted Voronoi-Delaunay on polyhedral surfaces.
This paper provides a stratification of semi-algebraic sets in the plane with finitely many geodesic segments.
problem How to stratify semi-algebraic sets in the plane with finitely many geodesic segments.
method Develops a semi-algebraic stratification of a real semi-algebraic set in the plane with open cells having the finiteness property.
result Provides insights for high-dimensional stratifications of semi-algebraic sets in connection with geodesics.
Hexagonal tilings minimize perimeter with unequal volumes.
problem Finding optimal tessellations with unequal cell volumes.
method Minimizing perimeter functionals for different classes of problems.
result Hexagonal tilings are optimal among partitions with almost equal areas.
Study classifies equidistant decompositions in 2D spaces.
problem Classifying equidistant decompositions in 2D spaces.
method Full classification of decompositions in Euclidean plane and sphere.
result Complete classification of equidistant decompositions in 2D spaces.
MREC efficiently matches and aligns point clouds, useful for single cell molecular data.
problem Comparing and aligning large datasets across various domains.
method Recursive decomposition algorithm for matching data sets, optimizing over partitioning and matching algorithms.
result Demonstrates flexibility and power in applying MREC to single cell molecular data alignment problems.
Proving geodesic triangulation spaces are Euclidean.
problem Proving spaces of geodesic triangulations are homeomorphic to Euclidean spaces.
method Proposing an approach to prove homeomorphism using negative curvature surfaces.
result Spaces of geodesic triangulations are homeomorphic to Euclidean spaces.
Develops methods to analyze feature-outcome associations in subpopulations.
problem Challenges in understanding feature-outcome associations in high-dimensional data.
method Geometric decomposition framework using gradient flow and co-monotonicity decomposition.
result Identifies context-dependent patterns and improves statistical power and interpretability.
A model for grid cells using vectors and matrices for position and motion.
problem Representing self-position and motion in a high-dimensional space.
method Vector-matrix multiplication, magnified local isometry, and global adjacency kernel.
result The model can learn hexagon patterns and correct errors.
This paper describes how to recover the topology of a closed manifold M from a good Morse function f on M. The essential method was suggested by Cohen, Jones and Segal. They constructed a topological category Cf and claimed that the classifying space BCf is homeomorphic to M. We prove it from a differ…
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these cells for the complement of a complex hyperplane arrangement defined over real …
The study refines contingency matrices for complex stratification and braid group cohomology.
problem Combinatorics of contingency matrices and their applications.
method Refinement of complex stratification and study of braid group cohomology.
result Totally positive meta-matrix formed by contingency matrix sizes.
We determine optimal inequalities for the systole of all hyperbolic compact surfaces of caracteristic -1. First, we study the geometry and topology of these surfaces. Then, we describe the action of modular groups on Teichmüller spaces. Finaly, we give cell decompositions of fundamental domains such as the set of systo…
The purpose of this paper is to describe certain natural 4-vector fields on quaternionic flag manifolds, which geometrically determine the Bruhat cell decomposition. This structure naturally descends from the symplectic group, where it is related to the dressing action given by the Iwasawa decomposition of the general …